Random signals

Histogram of the random signal

Continuous Time Sinusoidal signals

CT Signal (Periodic) A simple harmonic motion is described mathematically by

Complex Signal A complex exponential signal is

Discrete time sinusoidal signal

Discrete-time sinusoidal signal

Condition of Periodicity of DT signal

Effect of Sampling

Harmonically related complex exponential Continuous time exponentials

Discrete time exponentials Discrete time complex exponential is periodic if its relative frequency Is a rational number. We choose f 0 =1/N. We define a set of harmonically related complex exponentials by

Negative Frequencies

Analog to Digital Converter

ADC: To process analog signals by digital means, it is first necessary to convert them into digital form, that is, to convert them into a sequence of numbers having finite precision. The procedure is called analog to digital conversion and the devices used is called AD converters. 1.Sampling: This is the conversion of continuous time signal into discrete signal obtained by taking samples of continuous time signal at discrete time instants. If x a (t) the input to the sampler, the output x a (nT)=x(n) where T is the sampling interval. 2. Quantization: This is the conversion of discrete time continuous valued signal into a discrete time discrete valued (digital) signal. The value signal sample is selected from a finite set of possible values. The difference between the unquantized sample x(n) and the quantized output x q (n) is called quantization error. 3.In coding process each quantized output x q (n) is represented by a b-bit number. In speech processing the digital signal is converted back into analog form again. This process is called DA conversion. The DA conversion is done by connecting all the dots by some kind of interpolation.

The figure shows a form of DA conversion called stair-case approximation or zero-order hold. Other approximations are possible, connecting successive points linearly (linear interpolation) or fitting a quadratic through three successive points (quadratic approximation). For signals having finite frequency content (limited band width) sampling theorem specifies the optimum form of interpolation. Analog signal does not lose any information provided the sampling frequency is sufficiently high. Otherwise there is a problem called aliasing

Zero-order hold

Sampling of an analog signal